苏云飞;姚磊;

• 1:西北大学数学学院

摘要(Abstract)：

研究了二维空间中非齐次不可压缩Navier-Stokes/Vlasov-Fokker-Planck方程组的渐近分析,此模型用于塑造流体-粒子的相互作用.运用紧性方法得到ρ^ε,uε,u^ε的强收敛,最终得到由关于粒子宏观密度的对流-扩散方程及不可压缩Navier-Stokes方程组成的极限方程组.本文将相关文献的结果推广到非齐次不可压缩的情形.

关键词(KeyWords)： 非齐次不可压缩Navier-Stokes/Vlasov-Fokker-Planck方程组;弱解;渐近分析

Abstract:

Keywords:

基金项目(Foundation): 国家自然科学基金(11571280,11331005)

作者(Authors): 苏云飞;姚磊;

参考文献(References)：

• [1]Caflisch R,Papanicolaou G C.Dynamic theory of suspensions with Brownian effects[J].SIAM J.Appl.Math.,1983,43(4):885-906.
• [2]Sartory W K.Three-component analysis of blood sedimentation by the method of characteristics[J].Math.Biosci.,1997,33(1/2):145-165.
• [3]Williams F A.Spray Combustion and Atomization[J].Phys.of Fluids,1958,6(1):541-555.
• [4]Hamdache K.Global existence and large time behaviour of solutions for the Vlasov-Stokes equations[J].Janpan J.Indust.Appl.Math.,1998,15(1):51-74.
• [5]Boudin L,Desvillettes L.Grandmont C,et al.Global existence of solutions for the coupled Vlasov and Navier-Stokes equations[J].Differential Integral Equations,2009,22(11/12):1247-1271.
• [6]Chae M,Kang K,Lee J.Global existence of weak and classical solutions for the Navier-Stokes-VlasovFokker-Planck equations[J].J.Differential Equations,2011,251(9):2431-2465.
• [7]Wang D H,Yu C.Global weak solution to the inhomogeneous Navier-Stokes-Vlasov equations[J].J.Differential Equations,2015,259(8):3976-4008.
• [8]Yu C.Global weak solutions to the incompressible Navier-Stokes-Vlasov equations[J].J.Math.Pures Appl.,2013,100(2):275-293.
• [9]Mellet A,Vasseur A.Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations[J].Math.Models Methods Appl.Sci.,2007,17(7):1039-1063.
• [10]Baranger C,Desvillettes L.Coupling Euler and Vlasov equations in the context of sprays:the local-in-time,classical solutions[J].J.Hyperbolic Differ.Equ.,2006,3(1):1-26.
• [11]Carrillo J A,Choi Y P,Karper T K.On the analysis of a coupled kinetic-fluid model with local alignment forces[J].Ann.Inst.H.Poincar′e Anal.Non Lin′eaire,2016,33(2):273-307.
• [12]Goudon T.Asymptotic problems for a kinetic model of two-phase flow[J].Proc.Roy.Soc.Edinburgh Sect.A,2001,131(6):1371-1384.
• [13]Mellet A,Vasseur A.Asymptotic analysis for a Vlasov-Fokker-Planck/compressive Navier-Stokes system of equations[J].Comm.Math.Phys.,2008,281(3):573-596.
• [14]Goudon T,Jabin P E,Vasseur A.Hydrodynamic limit for the Vlasov-Navier-Stokes equations.I.Light particles regime[J].Indiana Univ.Math.J.,2004,53(6):1495-1515.
• [15]Goudon T,Jabin P E,Vasseur A.Hydrodynamic limit for the Vlasov-Navier-Stokes equations.II.Fine particles regime[J].Indiana Univ.Math.J.,2004,53(6):1517-1536.
• [16]Kang M J,Vasseur A.Asymptotic analysis of Vlasov-type equations under strong local alignment regime[J].Math.Models Methods Appl.Sci.,2015,25(11):2153-2173.
• [17]Figalli A,Kang M J.A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment[J].arxiv:1702.08087,2017.
• [18]Karper T K,Mellet A,Trivisa K.Hydrodynamic limit of the kinetic Cucker-Smale flocking model[J].Math.Models Methods Appl.Sci.,2015,25(1):131-163.
• [19]Lions P L.Mathematical Topics in Fluid Mechanics,Vol.II.Compressible Models[M].Oxford Lecture Series in Mathematics and its Applications.New York:The Clarendon Press,1998.
• [20]Lions P L.Mathematical Topics in Fluid Mechanics,Vol.I.Incompressible Models[M].Oxford Lecture Series in Mathematics and its Applications.New York:The Clarendon Press,1996.